No, it wouldn't be "Can I get one cup please", because that is incorrect. "Kudasai" literally means "Please give me", not "Can I get". If I am asking for something, I'm not going to say "Can I get" unless I want to go get it myself. If I'm in a restaurant and I want some water, I'm not going to say "Can I get some water" unless I want the server to point to the water dispenser. "Can I get" is an incorrect translation, and "Please give me" does not sound demanding at all. Besides, it is the accurate translation of "kudasai".

Same here, so I looked them up in jisho.org, according to which コップ is a glass (from Portuguese "copo"), and カップ is a cup. But that's just a dictionary; perhaps in real life, the line between the two blurs.

コップ = glass (drinking vessel); tumbler​, drinking glass, From Portuguese “copo”, From Dutch; Flemish “Kop”
cap isn't totaly wrong but the japanese word for cap is "カップ ". I f you want to hear "cap" please use this word. There are several ways to translate ~te kudasai. And I think "Can I get ...(please), is not the best one (my opinion). If you want to hear only that phrase, you should say so.

The phrase "コップを一つください" is a polite way to say "Please give me a cup." The English question "Can I get one cup?" Is unnatural to me, and indicates I want to get up retrieve the cup myself. "Kudasai" literally means "Please give me", and not "Can I get". This is prevalent throughout the Japanese course (which is otherwise excellent).